To determine the value of [tex]\(\sin 30^\circ\)[/tex], we need to recall some key properties of trigonometric functions and specific angles.
The sine function for common angles is often memorized or derived through knowledge of the unit circle or special triangles.
For the angle [tex]\(30^\circ\)[/tex]:
1. Unit Circle Approach: On the unit circle, [tex]\(30^\circ\)[/tex] corresponds to the point [tex]\( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \)[/tex]. Here, the y-coordinate gives the value of [tex]\(\sin 30^\circ\)[/tex].
2. Special Triangle Approach: Another method involves the 30-60-90 special right triangle. In this triangle, the sides have fixed ratios relative to each other. Specifically:
- The side opposite to the [tex]\(30^\circ\)[/tex] angle (the shorter leg) is [tex]\( \frac{1}{2} \)[/tex].
- The side opposite to the [tex]\(60^\circ\)[/tex] angle (the longer leg) is [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
- The hypotenuse is 1.
Given this information, the value of [tex]\(\sin 30^\circ\)[/tex] is the ratio of the length of the side opposite [tex]\(30^\circ\)[/tex] to the hypotenuse:
[tex]\[ \sin 30^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2} \][/tex]
Therefore, the correct answer is:
F. [tex]\(\frac{1}{2}\)[/tex]
